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Smooth Representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. Real or complex reductive Lie groups Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex Hilbert space ''V''I.e. a homomorphism (where GL(''V'') is the group of bounded linear operators on ''V'' whose inverse is also bounded and linear) such that the associated map is continuous. is called admissible if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''. An admissible representation π induces a (\mathfrak,K)-module which is easier to deal with as it is an algebraic obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Profinite Group
In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and the ''p''-adic Lie groups. Non-examples are real Lie groups, which have the no small subgroup property. In a locally profinite group, a closed subgroup is locally profinite, and every compact subgroup is contained in an open compact subgroup. Examples Important examples of locally profinite groups come from algebraic number theory. Let ''F'' be a non-archimedean local field. Then both ''F'' and F^\times are locally profinite. More generally, the matrix ring \operatorname_n(F) and the general linear group \operatorname_n( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gopal Prasad
Gopal Prasad (born 31 July 1945 in Ghazipur, India) is an Indian-American mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups. He is the Raoul Bott Professor of Mathematics at the University of Michigan in Ann Arbor. Education Prasad earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years later, in 1965, he received his master's degree in Mathematics from Patna University. After a brief stay at the Indian Institute of Technology Kanpur in their Ph.D. program for Mathematics, Prasad entered the Ph.D. program at the Tata Institute of Fundamental Research (TIFR) in 1966. There he began a long and extensive collaboration with his advisor M. S. Raghunathan on several topics including the study of lattices in semi-simple Lie groups and the congruence subgroup problem. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Evans Howe
Roger Evans Howe (born May 23, 1945) is the William R. Kenan, Jr. Professor Emeritus of Mathematics at Yale University, and Curtis D. Robert Endowed Chair in Mathematics Education at Texas A&M University. He is known for his contributions to representation theory, in particular for the notion of a reductive dual pair and the Howe correspondence, and his contributions to mathematics education. Biography He attended Ithaca High School, then Harvard University as an undergraduate, becoming a Putnam Fellow in 1964. He obtained his Ph.D. from University of California, Berkeley in 1969. His thesis, titled ''On representations of nilpotent groups'', was written under the supervision of Calvin Moore. Between 1969 and 1974, Howe taught at the State University of New York in Stony Brook before joining the Yale faculty in 1974. His doctoral students include Ju-Lee Kim, Jian-Shu Li, Zeev Rudnick, Eng-Chye Tan, and Chen-Bo Zhu. He moved to Texas A&M University in 2015. He has been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Zelevinsky
Andrei Vladlenovich Zelevinsky (; 30 January 1953 – 10 April 2013) was a Russian-American mathematician who made important contributions to algebra, combinatorics, and representation theory, among other areas. Biography Zelevinsky graduated in 1969 from the Moscow Mathematical School No. 2. After winning a silver medal as a member of the USSR team at the International Mathematical Olympiad he was admitted without examination to the mathematics department of Moscow State University where he obtained his PhD in 1978 under the mentorship of Joseph Bernstein, Alexandre Kirillov and Israel Gelfand. He worked in the mathematical laboratory of Vladimir Keilis-Borok at the Institute of Earth Science (1977–85), and at the Council for Cybernetics of the Soviet Academy of Sciences (1985–90). In the early 1980s, at a great personal risk, he taught at the Jewish People's University, an unofficial organization offering first-class mathematics education to talented students denied admiss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; he, יוס(י)ף נאומוביץ ברנשטיין; russian: Иосиф Наумович Бернштейн; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. Biography Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing anti-semitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). Awards and honors Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bill Casselman (mathematician)
William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a Professor Emeritus at the University of British Columbia. He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet. Career Casselman did his undergraduate work at Harvard College where his advisor was Raoul Bott and received his Ph.D from Princeton University in 1966 where his advisor was Goro Shimura. He was a visiting scholar at the Institute for Advanced Study in 1974, 1983, and 2001. He emigrated to Canada in 1971 and is a Professor Emeritus in mathematics at the University of British Columbia. Research Casselman specializes in representation theory, automorphic forms, geometric combinatorics, and the structure of algebraic groups. He has an interest in mathematical graphics and has been the graphics editor of the ''Notices of the American Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Locally Compact Group
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. Definition Let (''G'',''K'') be a pair consisting of a unimodular locally compact topological group ''G'' and a closed subgroup ''K'' of ''G''. Then the space of bi-''K''-invariant continuous functions of compact support :''C'' 'K''\''G''/''K'' can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted :''H''(''G''//''K'') and called the Hecke ring of the pair (''G'',''K''). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. Examples SL(2) In particular, this holds when :''G'' = ''SL''''n''(''Q''''p'') and ''K'' = ''SL''''n''(''Z''''p'') and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald. GL(2) On the other hand, in the case :''G'' = ''GL''2(Q) and ''K'' = ''GL''2(Z) we have the classical Hecke alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an algebraic curve over a finite field A function field of a variety is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adele Ring
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a record deal with XL Recordings. Her debut album, '' 19'', was released in 2008 and spawned the UK top-five singles "Chasing Pavements" and "Make You Feel My Love". The album was certified 8× platinum in the UK and triple platinum in the US. Adele was honoured with the Brit Award for Rising Star as well as the Grammy Award for Best New Artist. Adele released her second studio album, '' 21'', in 2011. It became the world's best-selling album of the 21st century, with sales of over 31 million copies. It was certified 18× platinum in the UK (the highest by a solo artist of all time) and Diamond in the US. According to ''Billboard'', ''21'' is the top-performing album in the US chart history, topping the ''Billboard'' 200 for 24 weeks (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
